In the literature there is another modification of the Hawking energy, due to Hayward . His suggestion is essentially with the only difference that the integrands above contain an additional term, namely the square of the anholonomicity (see Sections 4.1.8 and 11.2.1). However, we saw that is a boost gauge dependent quantity, thus the physical significance of this suggestion is questionable unless a natural boost gauge choice, e.g. in the form of a preferred foliation, is made. (Such a boost gauge might be that given by the main extrinsic curvature vector and discussed in Section 4.1.2.) Although the expression for the Hayward energy in terms of the GHP spin coefficients given in [63, 65] seems to be gauge invariant, this is due only to an implicit gauge choice. The correct, general GHP form of the extra term is . If, however, the GHP spinor dyad is fixed as in the large sphere or in the small sphere calculations, then , and hence the extra term is, in fact, .
Taking into account that near the future null infinity (see for example ), it is immediate from the remark on the asymptotic behaviour of above that the Hayward energy tends to the Newman-Unti instead of the Bondi-Sachs energy at the future null infinity. The Hayward energy has been calculated for small spheres both in non-vacuum and vacuum . In non-vacuum it gives the expected value . However, in vacuum it is , which is negative.
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